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'Proof Sorter - the Square Root of 2 Is Irrational' printed from http://nrich.maths.org/

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This method of proof can easily be generalised to prove that $\sqrt n$ is irrational when $n$ is not a square number .

What is the length of the diagonal of a square with sides of length 2?
How do we find the value of $\sqrt 2$?
What number has 2 as its square?
What is the side of a square which has area 2?

Now $(1.4)^2=1.96$, so the number $\sqrt 2$ is roughly $1.4$. To get a better approximation divide $2$ by $1.4$ giving about $1.428$, and take the average of $1.4$ and $1.428$ to get $1.414$. Repeating this process, $2\div 1.414 \approx 1.41443$ so $2\approx 1.414 \times 1.41443$, and the average of these gives the next approximation $1.414215$. We can continue this process indefinitely getting better approximations but never finding the square root exactly.

If $\sqrt 2$ were a rational number, that is if it could be written as a fraction $p/q$ where $p$ and $q$ are integers, then we could find the exact value. The proof sorter shows that this number is IRRATIONAL so we cannot find an exact value.